(May 5, 2015 at 2:53 am)Hatshepsut Wrote:(May 4, 2015 at 5:29 pm)Alex K Wrote: ...use riemannian geometry even if it's just SR. Otherwise I wouldn't know how one would correctly take the new coordinates into account.
I dare not clog the forum. Callaghan considers a special case. The non-rotating frame is presumed inertial. The two observers coincide in space and have no linear motion relative to one another. The rotation is uniform about the x-axis. The only world line tracked belongs to a point that remains stationary in the rotating frame. In this case, Callaghan just applied the classical rotation matrix to change the two space coordinates and a separate equation, T(t) = t * sqrt(1 - w^2*r^2/c^2) to find the proper time T on the world line, which is a helix in the non-rotating frame. It was simple because the point's radial distance r, common to both observers, and the rotating frame's angular rate w are constant. I can see this won't work if the point begins moving about. But I admit I'm in pretty heavy seas with this kind of stuff.
Nonetheless, though the point is stationary in the rotating frame, it's clock is slower relative to both observers the greater r is. Callagan also noted that the rotating frame has a boundary: it only covers events inside the cylinder of radius c/w centered on the x-axis. I agree it's a non-interesting situation: all this to describe a stationary dot as seen by two people, one of whom is dizzy!
So I've worked out the metric for a rotating frame, and it's interesting. There's no point where it stops being valid, but - as one would maybe expect - beyond the point you say, where w r > c, objects *have* to move clockwise with respect to the coordinates, and if you go further, they have to move fast than a certain speed with respect to the coordinates. This simply reflects the fact that when the observer rotates, objects far enough away all seem to rotate.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition